The Link between Equitable Partitions and Local Agreements in Multi-Agent Systems with Nonlinear Interactions

Nonlinear Analysis: Hybrid Systems

Authors

Anthony Couthures

Vineeth Satheeskumar Varma

Samson Lasaulce

Irinel-Constantin Morărescu

Published

August 2026

Abstract
Classically, global consensus is achieved in linear multi-agent systems that interact over a connected, unsigned graph. However, when the interactions are non-linear, agents may get polarized, i.e., they synchronize locally within communities while the communities do not reach a consensus with each other. In this context, we demonstrate that local synchronizations strongly rely on the existence of equitable partitions in the graph. Specifically, if some agents synchronize independently of the initial conditions, we prove that these agents must belong to the same cell of an equitable partition. On top of that, based on forward invariance properties, we are able to characterize the stability of local synchronization equilibria in terms of the stability of equilibria of a quotient graph defined by an equitable partition.

Paper PDF Slides PDF DOI

All publications

Citation

BibTeX citation:
@article{couthures2026,
  author = {Couthures, Anthony and Satheeskumar Varma, Vineeth and
    Lasaulce, Samson and Morărescu, Irinel-Constantin},
  title = {The {Link} Between {Equitable} {Partitions} and {Local}
    {Agreements} in {Multi-Agent} {Systems} with {Nonlinear}
    {Interactions}},
  journal = {Nonlinear Analysis: Hybrid Systems},
  volume = {61},
  pages = {101754},
  date = {2026-08},
  url = {https://anthonycouthures.github.io/publications/2026_nahs_equitable-partitions-local-agreements.html},
  doi = {10.1016/j.nahs.2026.101754},
  langid = {en},
  abstract = {Classically, global consensus is achieved in linear
    multi-agent systems that interact over a connected, unsigned graph.
    However, when the interactions are non-linear, agents may get
    polarized, i.e., they synchronize locally within communities while
    the communities do not reach a consensus with each other. In this
    context, we demonstrate that local synchronizations strongly rely on
    the existence of equitable partitions in the graph. Specifically, if
    some agents synchronize independently of the initial conditions, we
    prove that these agents must belong to the same cell of an equitable
    partition. On top of that, based on forward invariance properties,
    we are able to characterize the stability of local synchronization
    equilibria in terms of the stability of equilibria of a quotient
    graph defined by an equitable partition.}
}
For attribution, please cite this work as:
Anthony Couthures, Vineeth Satheeskumar Varma, Samson Lasaulce, and Irinel-Constantin Morărescu. 2026. “The Link between Equitable Partitions and Local Agreements in Multi-Agent Systems with Nonlinear Interactions.” Nonlinear Analysis: Hybrid Systems 61: 101754. https://doi.org/10.1016/j.nahs.2026.101754.